Coniveau 2 complete intersections and effective cones

The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture.